SIAM Conf. on Math for Industry, Oct. SIAM Conf. on Math for Industry, Oct. 10, 2009 10, 2009 Modeling Knots for Aesthetics and Simulations Carlo H. Séquin U.C. Berkeley Modeling, Analysis, Design …
Dec 21, 2015
SIAM Conf. on Math for Industry, Oct. 10, 2009SIAM Conf. on Math for Industry, Oct. 10, 2009
Modeling Knots for Aesthetics and Simulations
Carlo H. Séquin
U.C. Berkeley
Modeling, Analysis, Design …
Knots in Clothing Knots in Clothing
Knotted Appliances Knotted Appliances
Garden hose Power cable
Intricate Knots in the Realm of . . .Intricate Knots in the Realm of . . .
Boats Horses
Knots in ArtKnots in Art
Macrame Sculpture
Knotted PlantsKnotted Plants
Kelp Lianas
Knotted Building Blocks of LifeKnotted Building Blocks of Life
Knotted DNAModel of the most complex knotted protein (MIT 2006)
Mathematicians’ KnotsMathematicians’ Knots
Closed, non-self-intersecting curves in 3D space
0 3 4 6
Tabulated by their crossing-number :
= The minimal number of crossings visible after any deformation and projection
unknot
Various UnknotsVarious Unknots
3D Hilbert Curve (3D Hilbert Curve (SSééquin 2006quin 2006))
Pax Mundi II (Pax Mundi II (20072007))
Brent Collins, Steve Reinmuth, Carlo Séquin
The Simplest Real Knot: The TrefoilThe Simplest Real Knot: The Trefoil
José de Rivera, Construction #35
M. C. Escher, Knots (1965)
Complex, Symmetrical KnotsComplex, Symmetrical Knots
Tight “Braided” KnotsTight “Braided” Knots
Composite Knots Composite Knots
Knots can be “opened” at their periphery and then connected to each other.
Links and Linked KnotsLinks and Linked Knots
A link: comprises a set of loops
– possibly knotted and tangled together.
Two Linked Tori: Link 2Two Linked Tori: Link 22211
John Robinson, John Robinson, Bonds of Friendship (1979)Bonds of Friendship (1979)
Borromean Rings: Link 6Borromean Rings: Link 63322
John Robinson
Tetra Trefoil TanglesTetra Trefoil Tangles
Simple linking (1) -- Complex linking (2)
{over-over-under-under} {over-under-over-under}
Tetrahedral Trefoil Tangle Tetrahedral Trefoil Tangle (FDM)(FDM)
A Loose Tangle of TrefoilsA Loose Tangle of Trefoils
Dodecahedral Pentafoil ClusterDodecahedral Pentafoil Cluster
Realization: Extrude Hone - ProMetalRealization: Extrude Hone - ProMetal
Metal sintering and infiltration process
A Split TrefoilA Split Trefoil
To open: Rotate around z-axis
Split Trefoil (side view, closed)Split Trefoil (side view, closed)
Split Trefoil (side view, open)Split Trefoil (side view, open)
Splitting Moebius BandsSplitting Moebius Bands
Litho by FDM-model FDM-modelM.C.Escher thin, colored thick
Split Moebius Trefoil (SSplit Moebius Trefoil (Sééquin, 2003)quin, 2003)
““Knot DividedKnot Divided” Breckenridge, 2005” Breckenridge, 2005
Knotty ProblemKnotty Problem
How many crossings
does this “Not-Divided” Knot have ?
2.5D Celtic Knots – Basic Step2.5D Celtic Knots – Basic Step
Celtic Knot – Denser ConfigurationCeltic Knot – Denser Configuration
Celtic Knot – Second IterationCeltic Knot – Second Iteration
Recursive 9-Crossing KnotRecursive 9-Crossing Knot
Is this really a 81-crossing knot ?
9 crossings
Knot ClassificationKnot Classification
What kind of knot is this ?
Can you just look it up in the knot tables ?
How do you find a projection that yields the minimum number of crossings ?
There is still no completely safe method to assure that two knots are the same.
Project: “Beauty of Knots” Project: “Beauty of Knots”
Find maximal symmetry in 3D for simple knots.
Knot 41 and Knot 61
Computer Representation of KnotsComputer Representation of Knots
Spline representation via its control polygon.
String of piecewise-linear line segments.
But . . .
Is the Control Polygon Representative?Is the Control Polygon Representative?
A Problem:
You may construct a nice knotted control polygon,
and then find that the spline curve it defines
is not knotted at all !
Unknot With Knotted Control-PolygonUnknot With Knotted Control-Polygon
Composite of two cubic Bézier curves
Highly Knotted Control-PolygonsHighly Knotted Control-Polygons
Use the previous configuration as a building block.
Cut open lower left joint between the 2 Bézier segments.
Small changes will keep the control polygons knotted.
Assemble several such constructs in a cyclic compound.
Highly Knotted Control-PolygonsHighly Knotted Control-Polygons
The Result:
Control polygon has 12 crossings.
Compound Bézier curve is still the unknot!
An Intriguing Question:An Intriguing Question:
Can an un-knotted control polygon
produce a knotted spline curve ?
First guess: Probably NOT
Variation-diminishing property of Bézier curvesimplies that a spline cannot “wiggle”
more than its control polygon.
Cubic BCubic Béézier and Its Control Polygonzier and Its Control Polygon
Cubic Bézier curve
Region where curve is “outside” of control polygon
Two “entangled” curves
With “non-entangled” control polygons
Convex hull of control polygon
Two “Entangled” Bezier Segments “in 3D”Two “Entangled” Bezier Segments “in 3D”
NOTE: The 2 control polygons are NOT entangled!
The Building BlockThe Building Block
Two “entangled” curves
With “non-entangled” control polygons
Combining 4 such Entangled UnitsCombining 4 such Entangled Units
Use several units …
Control Polygons Are NOT Entangled …Control Polygons Are NOT Entangled …
Use several units …
Can Be Reduced to the ChordsCan Be Reduced to the Chords
This Is NOT a Knot !This Is NOT a Knot !
But This Is a Knot ! But This Is a Knot !
Knot 72
The ProblemThe Problem
When can we use the control polygon to make reliable predictions about the curve ?
Thus we have a true spline knotwhose control polygon is the unknot !
Tubular NeighborhoodsTubular Neighborhoods
A tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.
(Wikipedia)
( Tom Peters et al.)
Ambient IsotopyAmbient Isotopy
If both the curve and its control polygon lie in the same tubular neighborhood,they have the same topological surroundingsand thus have the same knotted-ness.
( Tom Peters et al.)
subdivided control polygon
A “Safe” Tubular NeighborhoodA “Safe” Tubular Neighborhood
A tube of uniform diameter equal to the minimum separation of any two branches
More Efficient Neighborhoods?More Efficient Neighborhoods?
Tube diameter is determined by tightest bottleneck
Inefficient!
Make tube diameter variable along the knot curve(s)
Difficult!
Another “Neighborhood”Another “Neighborhood”
The notion of the “control ribbon”:
control polygon
spline curve
controlribbon
A ruled surface, that connects points with equal parameter values on the spline and on the control polygon
Knots and Their Control RibbonsKnots and Their Control Ribbons
K31: “Trefoil” and K940: “Chinese Button Knot”
Crucial Test on Control RibbonCrucial Test on Control Ribbon
Any self-intersections ?
Does a Line Pass thru Control Ribbon?Does a Line Pass thru Control Ribbon?
Look at the “crossings”formed by close approaches betweenquery line (green)and the edges ofthe control ribbon.
If the two “crossings” have the same sign,line stabs the ribbon.
Current FocusCurrent Focus
Find out how this can be done most efficiently:
Find the occurrences of all “close approaches”
Determine the signs of the relevant “crossings”
ConclusionConclusion
Knots appear in many domains, in many different forms, and with highly varying degrees of complexity.
CAD tools have only tangentially addressed efficient modeling and analysis of knotted structures.
Suitable abstractions of knots, coupled with some topological guarantees, offer promise for computationally efficient solutions.
The “quest” has only just begun!
AcknowledgementsAcknowledgements
Thanks to Tom Peters for many fruitful discussions!
This work is being supported in part by the Center for Hybrid and Embedded Software Systems (CHESS) at UC Berkeley, which receives support from the National Science Foundation (NSF award #CCR-0225610 (ITR)).
Q U E S T I O N S ?Q U E S T I O N S ?
Granny-Knot-Lattice (SGranny-Knot-Lattice (Séquin, 1981)quin, 1981)